# What is Inner product: Definition and 310 Discussions

In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets (as in

a
,
b

{\displaystyle \langle a,b\rangle }
). Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.An inner product naturally induces an associated norm, (

|

x

|

{\displaystyle |x|}
and

|

y

|

{\displaystyle |y|}
are the norms of

x

{\displaystyle x}
and

y

{\displaystyle y}
in the picture), which canonically makes every inner product space into a normed vector space. If this normed space is also a Banach space then the inner product space is called a Hilbert space. If an inner product space

(
H
,

,

)

{\displaystyle (H,\langle \,\cdot \,,\,\cdot \,\rangle )}
is not a Hilbert space then it can be "extended" to a Hilbert space

(

H
¯

,

,

H
¯

)

,

{\displaystyle \left({\overline {H}},\langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}\right),}
called a completion. Explicitly, this means that

H

{\displaystyle H}
is linearly and isometrically embedded onto a dense vector subspace of

H
¯

{\displaystyle {\overline {H}}}
and that the inner product

,

H
¯

{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}}
on

H
¯

{\displaystyle {\overline {H}}}
is the unique continuous extension of the original inner product

,

{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }
.

View More On Wikipedia.org
1. ### POTW A Modified Basis in an Inner Product Space

Given an orthonormal basis ##\{e_1,\ldots, e_n\}## in a complex inner product space ##V## of dimension ##n##, show that if ##v_1,\ldots, v_n\in V## such that ##\sum_{j = 1}^n \|v_j\|^2 < 1##, then ##\{v_1 + e_1,\ldots, v_n + e_n\}## is a basis for ##V##.
2. ### I Inner product - positive or positive semidefinite?

Hi In QM the inner product satisfies < a | a > ≥ 0 with equality if and only if a = 0. Is this positive definite or positive semidefinite because i have seen it described as both Thanks

25. ### I Inner product of a vector with an operator

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...
26. ### I Complex conjugate of an inner product

Hi everyone. Yesterday I had an exam, and I spent half the exam trying to solve this question. Show that ##\left\langle\Psi\left(\vec{r}\right)\right|\hat{p_{y}^{2}}\left|\phi\left(\vec{r}\right)\right\rangle =\left\langle...
27. ### I Description of Inner Product

Hey, I am currently reading over the linear algebra section of the "introduction to quantum mechanics" by Griffiths, in the Inner product he notes: "The inner product of two vector can be written very neatly in terms of their components: <a|B>=a1* B1 + a2* B ... " He also took upon the...
28. ### I Computing inner products of spherical harmonics

In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a...
29. ### Convex Set in R^n Problem

Homework Statement Let ##C \subset \mathbb{R}^n## a convex set. If ##x \in \mathbb{R}^n## and ##\overline{x} \in C## are points that satisfy ##|x-\overline{x}|=d(x,C)##, proves that ##\langle x-\overline{x},y-\overline{x} \rangle \leq 0## for all ##y \in C##. Homework Equations By definition...
30. ### Orthogonal Vectors in Rn Problem

Homework Statement Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp (b-a)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##|c|<|x|##. Homework Equations The first...
31. ### Inner product - Analysis in Rn problem

Homework Statement Let ##x,y \in \mathbb{R^n}## not null vectors. If for all ##z \in \mathbb{R^n}## that is orthogonal to ##x## we have that ##z## is also orthogonal to ##y##, prove that ##x## and ##y## are multiple of each other. Homework Equations We can use that fact that ##<x ...
32. ### I Real function inner product space

Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is...
33. ### Expectation value of raising and lower operator

I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result. Homework Statement At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by: ## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...
34. ### I Basis Vectors & Inner Product: A No-Nonsense Introduction

I read from this page https://properphysics.wordpress.com/2014/06/09/a-no-nonsense-introduction-to-special-relativity-part-6/ that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...
35. ### I Normalized basis when taking inner product

Consider that a vector can be represented in two different basis. My question is do we need to normalize both basis before taking the inner product? What motivates this question is because I found out that the inner product of a vector having components ##a,b## in the normalized polar basis of...
36. ### Inner Product, Triangle and Cauchy Schwarz Inequalities

Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
37. ### Insights Hilbert Spaces and Their Relatives - Comments

Greg Bernhardt submitted a new PF Insights post Hilbert Spaces and Their Relatives Continue reading the Original PF Insights Post.
38. ### Laplace expansion of the inner product (Geometric Algebra)

Homework Statement Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r} - \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##. Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an r-blade with ##r > 0##. Also, the dot...
39. ### A I'm getting the wrong inner product of Fock space

I am trying to follow modern QFT by Tom Banks and I am having an issue with literally the first equation. He claims that beginning from ## |p_1 , p_2, ... , p_k> \: = \: a^\dagger (p_1) a^\dagger (p_2) \cdots a^\dagger (p_k)|0> ## with the commutation relation ##[a (p),a^\dagger (q)]_\pm \: =...
40. S

### I Eigenvectors and inner product

Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix? I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete. Can two eigenvectors which...
41. S

### A Eigenvectors and matrix inner product

Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula : $$\int x(t)\overline y(t) dt$$ on the x and y coordinates of the eigenvectors [x_1,y_1] and...
42. ### I Define inner product of vector fields EM

I'm reading a textbook on electromagnetism. It says that for two vector fields ##\textbf{F}(\textbf{r})## and ##\textbf{G}(\textbf{r})## their inner product is defined as ##(\textbf{F},\textbf{G}) = \int \textbf{F}^{*}\cdot \textbf{G} \thinspace d^3\textbf{r}## And that if ##\textbf{F}## is...
43. ### B Inner product of functions of continuous variable

I am new to quantum mechanics and I have recently been reading Shankar's book. It was all good until I reached the idea of representing functions of continouis variable as kets for example |f(x)>. The book just scraped off the definition of inner product in the discrete space case and refined it...
44. P

### Show this integral defines a scalar product.

Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
45. ### I Inner product, dot product?

I started learning quantum, and I got a bit confused about inner and dot products. I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online. Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate. The PDF from MIT...
46. O

### Inner Product Question

Homework Statement Homework Equations The Attempt at a Solution [/B] I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one. Follow all the procedures I already did:
47. ### I Non-negativity of the inner product

The inner product axioms are the following: ##\text{(a)} \ \langle x+z,y \rangle = \langle x,y \rangle + \langle z,y \rangle## ##\text{(b)} \ \langle cx,y \rangle = c\langle x,y \rangle## ##\text{(c)} \ \overline{\langle x,y \rangle} = \langle y,x \rangle## ##\text{(d)} \ \langle x,x \rangle > 0...
48. ### Proving a fact about inner product spaces

Homework Statement Let ##V## be a vector space equipped with an inner product ##\langle \cdot, \cdot \rangle##. If ##\langle x,y \rangle = \langle x, z\rangle## for all ##x \in V##, then ##y=z##. Homework Equations The Attempt at a Solution Here is my attempt. ##\langle x,y \rangle = \langle...
49. ### Eigenvectors and orthogonal basis

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
50. ### Proving a function is an inner product in a complex space

Homework Statement Prove the following form for an inner product in a complex space V: ##\langle u,v \rangle## ##=## ##\frac 1 4####\left| u+v\right|^2## ##-## ##\frac 1 4####\left| u-v\right|^2## ##+## ##\frac 1 4####\left| u+iv\right|^2## ##-## ##\frac 1 4####\left| u-iv\right|^2## Homework...